What is expand (e^xcos(x)-e^xsin(x))^2 ?

The solution to expand (e^xcos(x)-e^xsin(x))^2 is e^{2x}(-sin(2x)+1)

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expand (e^xcos(x)-e^xsin(x))^2

expand

(e^{x} cos (x) - e^{x} sin (x))^{2}

e^{2 x} (- sin (2 x) + 1)

Solution steps

(e^{x} cos (x) - e^{x} sin (x))^{2}

Apply Perfect Square Formula:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = e^{x} cos (x), b = e^{x} sin (x)

= (e^{x} cos (x))^{2} - 2 e^{x} cos (x) e^{x} sin (x) + (e^{x} sin (x))^{2}

Simplify

(e^{x} cos (x))^{2} - 2 e^{x} cos (x) e^{x} sin (x) + (e^{x} sin (x))^{2} : e^{2 x} cos^{2} (x) - 2 e^{2 x} cos (x) sin (x) + e^{2 x} sin^{2} (x)

= e^{2 x} cos^{2} (x) - 2 e^{2 x} cos (x) sin (x) + e^{2 x} sin^{2} (x)

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

= cos^{2} (x) e^{2 x} + sin^{2} (x) e^{2 x} - sin (2 x) e^{2 x}

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

= e^{2 x} (- sin (2 x) + 1)

tan^{2} (a) (2 cos^{2} (a) + sin^{2} (a) - 1)

(1 + x)^{2}

(2 (n + 1))!

(x + 3) (x - 3)

cot (x) (sec (x) - tan (x))

What is expand (e^xcos(x)-e^xsin(x))^2 ?
The solution to expand (e^xcos(x)-e^xsin(x))^2 is e^{2x}(-sin(2x)+1)